This is the first book which explicitly uses Mathematica (computer algebra system) to allow researchers and students to more easily compute and solve almost any kind of differential equation using Lie"s theory. Heretofore time-consuming and cumbersome calculations if done by hand, are much more easily and quickly performed via the Mathematica computer algebra software. The material in this book, and on the accompanying CD-ROM, should be of interest to a broad group of scientists, mathematicians and engineers involved in dealing with symmetry analysis of differential equations. Each section of the book starts with a theoretical discussion of the material, then shows the application in connection with Mathematica. This book contains a large number of working examples relating to these applications of Lie"s theory. The cross-platform CD-ROM contains Mathematica (version 3.0) notebooks which provide users with the capability of directly interacting with the code presented within the book. In addition, the author"s proprietary "MathLie" software is included, so users can readily learn to use this powerful tool to perform algebraic computations.